[@notconfusing‽](http://twitter.com/notconfusing)
from IPython import display
display.Image('sets0.png')
display.Image('sets01.png')
According to his definition:
a set \(S\) is any collection of definite, distinguishable objects of our intution or of our intellect to be conceived as a whole. The objects are called the elements or members of \(S\)
display.Image('sets1.png')
display.Image('sets2.png')
We have to able to say if two elements are the same.
We have to be able to say definitely if a member is, or is not a member of a set.
\(x \in S\)
"x is an element of S"
or
\(x \notin S\)
"x is not an element of S"
\(S = \{1, 2, 3, 14\}\)
"S is the set with elements 1, and 2, 3, and fourteen"
\(S = \{x | x \text{ is even} \}\)
"The set of all numbers such that those numbers are even."
"The set of all even numbers"
\(S = \{\text{horse} | \text{horse has three legs} \}\)
"The set of all horses such that the horse has three legs".
"The set of all three-legged horeses."
Is \(\{2, 4, 6\} = \{2, 4, 4, 6\}\)?
Is \(\{2, 4, 6\} = \{6, 2, 4\}\)?
Is \(\{\text{Morrissey}, \text{Bowie}\} = \{\{\text{Morrissey}, \text{Bowie}\}\} = \{\{\text{Bowie}\}, \{\text{Morissey}\}\}\) ?
The set that contains no elements is the empty set.
Looks like: \(\emptyset\)
Question is \(\emptyset = \{\emptyset\}\)?
No, but why?
Let's just think about the size of each set. On the left the size is zero, on the right the size is one.
display.Image('sets3.png')
A big Math scandal, which temporarily destroyed set theory. I'll give my own reformulation as the Yoga Teacher Paradox, since the original version was needlessly male-oriented:
Question: does the Yoga teacher, teach themselves?
Set theory recovered from this problem by the creation of classes which are different from sets. But that's for another day.
Hopefully you get the idea that sets are basically minimal rigour to our basic understanding of what a collection of object is. It's so fundamental you actually don't need any Math for it, and in fact numbers are built on Set Theory.
Your homework is to use the concept of "empty set" in dinner time conversation.